Rank 4 stable vector bundles on hyperk\"ahler fourfolds of Kummer type

We partially extend to hyperkähler fourfolds of Kummer type the results on hyperkähler (HK) varieties of type K3[2] by the author in a 2022 paper (and later extended to HK varieties of type K3[n]). Let (M,h) be a general polarized HK fourfold of Kummer type such that qM(h) ≡ −6 (mod 16) and div(h) = 2, or qM(h) ≡ −6 (mod 144) and div(h) = 6. We show that there exists a unique (up to isomorphism) slope stable vector bundle F on M such that r(F) = 4, c1(F) = h, ∆(F) = c2(M). Moreover, F is rigid. One of our motivations is the desire to describe explicitly a locally complete family of polarized HK fourfolds of Kummer type.